The graded structure of algebraic Cuntz-Pimsner rings

Abstract

Algebraic Cuntz-Pimsner rings are naturally Z-graded rings that generalize corner skew Laurent polynomial rings, Leavitt path algebras and unperforated Z-graded Steinberg algebras. In this article, we characterize strongly, epsilon-strongly and nearly epsilonstrongly Z-graded algebraic Cuntz-Pimsner rings up to graded isomorphism. We recover two results by Hazrat on when corner skew Laurent polynomial rings and Leavitt path algebras are strongly graded. As a further application, we characterize noetherian and artinian corner skew Laurent polynomial rings.

Prime group graded rings with applications to partial crossed products and Leavitt path algebras

“…Moreover, unital partial crossed products (see [32]), Leavitt path algebras of finite graphs (see [31]), and certain Cuntz-Pimsner rings (see [24]) are classes of graded rings that are epsilon-strongly graded. A further generalization was introduced by Nystedt and Öinert: Definition 2.10 ([31, Def.…”

“…Indeed for v 1 , v 2 we have v 1 ≥ v 2 and v 2 ≥ v 2 . A computation yields L R (E 3 ) ∼ = M 2 (R) (see [24,Expl. 2.6]).…”

Chain Conditions for Epsilon-Strongly Graded Rings with Applications to Leavitt Path Algebras

“…We consider the special case of a fractional skew monoid ring by a corner isomorphism which is also called a corner skew Laurent polynomial ring. Let R be a unital ring and let α : R → eRe be a corner ring isomorphism where e is an idempotent of R. The corner skew Laurent polynomial ring, denoted by R[t + , t − ; α], is a unital epsilon-strongly Z-graded ring (see [20,Prop. 8.1]).…”

“…This class of graded rings includes: unital partial crossed products (see [24, pg. 2]), corner skew Laurent polynomial rings (see [20,Thm. 8.1]) and Leavitt path algebras of finite graphs (see [23,Thm.…”

On the structure of nearly epsilon and epsilon-strongly graded rings

“…On the other hand, partial actions of groups have been introduced in the theory of operator algebras giving powerful tools of their study, and in a pure algebraic context were first studied in [7], later the possibility to construct a crossed product based on a partial action suggested the idea of creating a corresponding Galois Theory [8]. Crossed products related to partial actions are graded rings which are not necessarily strong, but belong to a more general class, the so called epsilon-strongly graded rings (see Definition 3.5), this class was recently introduced in [17] and has being a subject of increasingly study (see [11,12,13,16,17,18]). Relevant families of rings which can be endowed with an epsilon-strong gradation include Morita rings, Leavitt Path Algebras associated to finite graphs, crossed product by partial actions and corner skew polynomial rings.…”